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.\"     from: @(#)exp.3	6.12 (Berkeley) 7/31/91
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.Dd $Mdocdate: January 15 2015 $
.Dt EXP 3
.Os
.Sh NAME
.Nm exp ,
.Nm expf ,
.Nm expl ,
.Nm exp2 ,
.Nm exp2f ,
.Nm exp2l ,
.Nm expm1 ,
.Nm expm1f ,
.Nm expm1l ,
.Nm log ,
.Nm logf ,
.Nm logl ,
.Nm log2 ,
.Nm log2f ,
.Nm log2l ,
.Nm log10 ,
.Nm log10f ,
.Nm log10l ,
.Nm log1p ,
.Nm log1pf ,
.Nm log1pl ,
.Nm pow ,
.Nm powf ,
.Nm powl
.Nd exponential, logarithm, power functions
.Sh SYNOPSIS
.In math.h
.Ft double
.Fn exp "double x"
.Ft float
.Fn expf "float x"
.Ft long double
.Fn expl "long double x"
.Ft double
.Fn exp2 "double x"
.Ft float
.Fn exp2f "float x"
.Ft long double
.Fn exp2l "long double x"
.Ft double
.Fn expm1 "double x"
.Ft float
.Fn expm1f "float x"
.Ft long double
.Fn expm1l "long double x"
.Ft double
.Fn log "double x"
.Ft float
.Fn logf "float x"
.Ft long double
.Fn logl "long double x"
.Ft double
.Fn log2 "double x"
.Ft float
.Fn log2f "float x"
.Ft long double
.Fn log2l "long double x"
.Ft double
.Fn log10 "double x"
.Ft float
.Fn log10f "float x"
.Ft long double
.Fn log10l "long double x"
.Ft double
.Fn log1p "double x"
.Ft float
.Fn log1pf "float x"
.Ft long double
.Fn log1pl "long double x"
.Ft double
.Fn pow "double x" "double y"
.Ft float
.Fn powf "float x" "float y"
.Ft long double
.Fn powl "long double x" "long double y"
.Sh DESCRIPTION
The
.Fn exp
function computes the base
.Ms e
exponential value of the given argument
.Fa x .
The
.Fn expf
function is a single precision version of
.Fn exp .
The
.Fn expl
function is an extended precision version of
.Fn exp .
.Pp
The
.Fn exp2
function computes the base 2 exponential of the given argument
.Fa x .
The
.Fn exp2f
function is a single precision version of
.Fn exp2 .
The
.Fn exp2l
function is an extended precision version of
.Fn exp2 .
.Pp
The
.Fn expm1
function computes the value exp(x)\-1 accurately even for tiny argument
.Fa x .
The
.Fn expm1f
function is a single precision version of
.Fn expm1 .
The
.Fn expm1l
function is an extended precision version of
.Fn expm1 .
.Pp
The
.Fn log
function computes the value of the natural logarithm of argument
.Fa x .
The
.Fn logf
function is a single precision version of
.Fn log .
The
.Fn logl
function is an extended precision version of
.Fn log .
.Pp
The
.Fn log2
function computes the value of the logarithm of argument
.Fa x
to base 2.
The
.Fn log2f
function is a single precision version of
.Fn log2 .
The
.Fn log2l
function is an extended precision version of
.Fn log2 .
.Pp
The
.Fn log10
function computes the value of the logarithm of argument
.Fa x
to base 10.
The
.Fn log10f
function is a single precision version of
.Fn log10 .
The
.Fn log10l
function is an extended precision version of
.Fn log10 .
.Pp
The
.Fn log1p
function computes
the value of log(1+x) accurately even for tiny argument
.Fa x .
The
.Fn log1pf
function is a single precision version of
.Fn log1p .
The
.Fn log1pl
function is an extended precision version of
.Fn log1p .
.Pp
The
.Fn pow
function computes the value of
.Ar x
to the exponent
.Ar y .
The
.Fn powf
function is a single precision version of
.Fn pow .
The
.Fn powl
function is an extended precision version of
.Fn pow .
.Sh RETURN VALUES
These functions will return the appropriate computation unless an error
occurs or an argument is out of range.
The functions
.Fn exp ,
.Fn expm1
and
.Fn pow
detect if the computed value will overflow,
set the global variable
.Va errno
to
.Er ERANGE
and cause a reserved operand fault on a VAX or Tahoe.
The function
.Fn pow x y
checks to see if
.Fa x
< 0 and
.Fa y
is not an integer, in the event this is true,
the global variable
.Va errno
is set to
.Er EDOM
and on the VAX and Tahoe generate a reserved operand fault.
On a VAX and Tahoe,
.Va errno
is set to
.Er EDOM
and the reserved operand is returned
by log unless
.Fa x
> 0, by
.Fn log1p
unless
.Fa x
> \-1.
.Sh ERRORS (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within
an
.Em ulp ,
and log10(x) to within about 2
.Em ulps ;
an
.Em ulp
is one
.Em Unit
in the
.Em Last
.Em Place .
The error in
.Fn pow x y
is below about 2
.Em ulps
when its
magnitude is moderate, but increases as
.Fn pow x y
approaches
the over/underflow thresholds until almost as many bits could be
lost as are occupied by the floating\-point format's exponent
field; that is 8 bits for
.Dq VAX D
and 11 bits for IEEE 754 Double.
No such drastic loss has been exposed by testing; the worst
errors observed have been below 20
.Em ulps
for
.Dq VAX D ,
300
.Em ulps
for IEEE 754 Double.
Moderate values of
.Fn pow
are accurate enough that
.Fn pow integer integer
is exact until it is bigger than 2**56 on a VAX,
2**53 for IEEE 754.
.Sh NOTES
The functions exp(x)\-1 and log(1+x) are called
expm1 and logp1 in BASIC on the Hewlett\-Packard HP-71B
and APPLE Macintosh, EXP1 and LN1 in Pascal, exp1 and log1 in C
on APPLE Macintoshes, where they have been provided to make
sure financial calculations of ((1+x)**n\-1)/x, namely
expm1(n\(**log1p(x))/x, will be accurate when x is tiny.
They also provide accurate inverse hyperbolic functions.
.Pp
The function
.Fn pow x 0
returns x**0 = 1 for all x including x = 0,
.if n \
\*(If
.if t \
\(if
(not found on a VAX),
and
.Em \*(Na
(the reserved operand on a VAX).
Previous implementations of
.Fn pow
may have defined x**0 to be undefined in some or all of these cases.
Here are reasons for returning x**0 = 1 always:
.Bl -enum -width indent
.It
Any program that already tests whether x is zero (or
infinite or \*(Na) before computing x**0 cannot care
whether 0**0 = 1 or not.
Any program that depends upon 0**0 to be invalid is dubious anyway since that
expression's meaning and, if invalid, its consequences
vary from one computer system to another.
.It
Some Algebra texts (e.g., Sigler's) define x**0 = 1 for
all x, including x = 0.
This is compatible with the convention that accepts a[0]
as the value of polynomial
.Bd -literal -offset indent
p(x) = a[0]\(**x**0 + a[1]\(**x**1 + a[2]\(**x**2 +...+ a[n]\(**x**n
.Ed
.Pp
at x = 0 rather than reject a[0]\(**0**0 as invalid.
.It
Analysts will accept 0**0 = 1 despite that x**y can
approach anything or nothing as x and y approach 0
independently.
The reason for setting 0**0 = 1 anyway is this:
.Bd -filled -offset indent
If x(z) and y(z) are
.Em any
functions analytic (expandable
in power series) in z around z = 0, and if there
x(0) = y(0) = 0, then x(z)**y(z) \(-> 1 as z \(-> 0.
.Ed
.It
If 0**0 = 1, then
.if n \
\*(If**0 = 1/0**0 = 1 too; and
.if t \
\(if**0 = 1/0**0 = 1 too; and
then \*(Na**0 = 1 too because x**0 = 1 for all finite
and infinite x, i.e., independently of x.
.El
.Sh SEE ALSO
.Xr ilogb 3 ,
.Xr infnan 3
.Sh HISTORY
The
.Fn exp
and
.Fn log
functions first appeared in
.At v1 ;
.Fn pow
in
.At v3 ;
.Fn log10
in
.At v7 ;
.Fn log1p
and
.Fn expm1
in
.Bx 4.3 .
